3.8.16 \(\int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^5}{(a+i a \tan (e+f x))^2} \, dx\) [716]

Optimal. Leaf size=194 \[ \frac {8 (3 A+7 i B) c^5 x}{a^2}+\frac {8 (3 i A-7 B) c^5 \log (\cos (e+f x))}{a^2 f}-\frac {8 (i A-B) c^5}{a^2 f (i-\tan (e+f x))^2}+\frac {16 (2 A+3 i B) c^5}{a^2 f (i-\tan (e+f x))}-\frac {(7 A+24 i B) c^5 \tan (e+f x)}{a^2 f}+\frac {(i A-7 B) c^5 \tan ^2(e+f x)}{2 a^2 f}+\frac {i B c^5 \tan ^3(e+f x)}{3 a^2 f} \]

[Out]

8*(3*A+7*I*B)*c^5*x/a^2+8*(3*I*A-7*B)*c^5*ln(cos(f*x+e))/a^2/f-8*(I*A-B)*c^5/a^2/f/(I-tan(f*x+e))^2+16*(2*A+3*
I*B)*c^5/a^2/f/(I-tan(f*x+e))-(7*A+24*I*B)*c^5*tan(f*x+e)/a^2/f+1/2*(I*A-7*B)*c^5*tan(f*x+e)^2/a^2/f+1/3*I*B*c
^5*tan(f*x+e)^3/a^2/f

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Rubi [A]
time = 0.17, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {3669, 78} \begin {gather*} \frac {c^5 (-7 B+i A) \tan ^2(e+f x)}{2 a^2 f}-\frac {c^5 (7 A+24 i B) \tan (e+f x)}{a^2 f}+\frac {16 c^5 (2 A+3 i B)}{a^2 f (-\tan (e+f x)+i)}-\frac {8 c^5 (-B+i A)}{a^2 f (-\tan (e+f x)+i)^2}+\frac {8 c^5 (-7 B+3 i A) \log (\cos (e+f x))}{a^2 f}+\frac {8 c^5 x (3 A+7 i B)}{a^2}+\frac {i B c^5 \tan ^3(e+f x)}{3 a^2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^5)/(a + I*a*Tan[e + f*x])^2,x]

[Out]

(8*(3*A + (7*I)*B)*c^5*x)/a^2 + (8*((3*I)*A - 7*B)*c^5*Log[Cos[e + f*x]])/(a^2*f) - (8*(I*A - B)*c^5)/(a^2*f*(
I - Tan[e + f*x])^2) + (16*(2*A + (3*I)*B)*c^5)/(a^2*f*(I - Tan[e + f*x])) - ((7*A + (24*I)*B)*c^5*Tan[e + f*x
])/(a^2*f) + ((I*A - 7*B)*c^5*Tan[e + f*x]^2)/(2*a^2*f) + ((I/3)*B*c^5*Tan[e + f*x]^3)/(a^2*f)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 3669

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a*(c/f), Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1)*(A + B*x), x
], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]

Rubi steps

\begin {align*} \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^5}{(a+i a \tan (e+f x))^2} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {(A+B x) (c-i c x)^4}{(a+i a x)^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \text {Subst}\left (\int \left (-\frac {(7 A+24 i B) c^4}{a^3}+\frac {i (A+7 i B) c^4 x}{a^3}+\frac {i B c^4 x^2}{a^3}+\frac {16 i (A+i B) c^4}{a^3 (-i+x)^3}+\frac {16 (2 A+3 i B) c^4}{a^3 (-i+x)^2}+\frac {8 (-3 i A+7 B) c^4}{a^3 (-i+x)}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {8 (3 A+7 i B) c^5 x}{a^2}+\frac {8 (3 i A-7 B) c^5 \log (\cos (e+f x))}{a^2 f}-\frac {8 (i A-B) c^5}{a^2 f (i-\tan (e+f x))^2}+\frac {16 (2 A+3 i B) c^5}{a^2 f (i-\tan (e+f x))}-\frac {(7 A+24 i B) c^5 \tan (e+f x)}{a^2 f}+\frac {(i A-7 B) c^5 \tan ^2(e+f x)}{2 a^2 f}+\frac {i B c^5 \tan ^3(e+f x)}{3 a^2 f}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1357\) vs. \(2(194)=388\).
time = 7.53, size = 1357, normalized size = 6.99 \begin {gather*} \frac {4 (-3 i A+5 B) c^5 \cos (2 f x) \sec (e+f x) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x))}{f (A \cos (e+f x)+B \sin (e+f x)) (a+i a \tan (e+f x))^2}+\frac {\sec (e+f x) \left (3 A c^5 \cos (e)+7 i B c^5 \cos (e)+3 i A c^5 \sin (e)-7 B c^5 \sin (e)\right ) (8 \text {ArcTan}(\tan (f x)) \cos (e)+8 i \text {ArcTan}(\tan (f x)) \sin (e)) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x))}{f (A \cos (e+f x)+B \sin (e+f x)) (a+i a \tan (e+f x))^2}+\frac {\sec (e+f x) \left (3 A c^5 \cos (e)+7 i B c^5 \cos (e)+3 i A c^5 \sin (e)-7 B c^5 \sin (e)\right ) \left (4 i \cos (e) \log \left (\cos ^2(e+f x)\right )-4 \log \left (\cos ^2(e+f x)\right ) \sin (e)\right ) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x))}{f (A \cos (e+f x)+B \sin (e+f x)) (a+i a \tan (e+f x))^2}+\frac {\sec (e) \sec ^3(e+f x) (3 A \cos (e)+21 i B \cos (e)+2 B \sin (e)) \left (\frac {1}{6} i c^5 \cos (2 e)-\frac {1}{6} c^5 \sin (2 e)\right ) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x))}{f (A \cos (e+f x)+B \sin (e+f x)) (a+i a \tan (e+f x))^2}+\frac {(A+i B) \cos (4 f x) \sec (e+f x) \left (2 i c^5 \cos (2 e)+2 c^5 \sin (2 e)\right ) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x))}{f (A \cos (e+f x)+B \sin (e+f x)) (a+i a \tan (e+f x))^2}+\frac {(3 A+7 i B) \sec (e+f x) \left (8 c^5 f x \cos (2 e)+8 i c^5 f x \sin (2 e)\right ) (\cos (f x)+i \sin (f x))^2 (A+B \tan (e+f x))}{f (A \cos (e+f x)+B \sin (e+f x)) (a+i a \tan (e+f x))^2}-\frac {4 (3 A+5 i B) c^5 \sec (e+f x) (\cos (f x)+i \sin (f x))^2 \sin (2 f x) (A+B \tan (e+f x))}{f (A \cos (e+f x)+B \sin (e+f x)) (a+i a \tan (e+f x))^2}+\frac {(A+i B) \sec (e+f x) \left (2 c^5 \cos (2 e)-2 i c^5 \sin (2 e)\right ) (\cos (f x)+i \sin (f x))^2 \sin (4 f x) (A+B \tan (e+f x))}{f (A \cos (e+f x)+B \sin (e+f x)) (a+i a \tan (e+f x))^2}+\frac {\sec (e) \sec ^4(e+f x) (\cos (f x)+i \sin (f x))^2 \left (-\frac {1}{2} B c^5 \cos (2 e-f x)+\frac {1}{2} B c^5 \cos (2 e+f x)-\frac {1}{2} i B c^5 \sin (2 e-f x)+\frac {1}{2} i B c^5 \sin (2 e+f x)\right ) (A+B \tan (e+f x))}{3 f (A \cos (e+f x)+B \sin (e+f x)) (a+i a \tan (e+f x))^2}+\frac {\sec (e) \sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \left (-\frac {21}{2} i A c^5 \cos (2 e-f x)+\frac {73}{2} B c^5 \cos (2 e-f x)+\frac {21}{2} i A c^5 \cos (2 e+f x)-\frac {73}{2} B c^5 \cos (2 e+f x)+\frac {21}{2} A c^5 \sin (2 e-f x)+\frac {73}{2} i B c^5 \sin (2 e-f x)-\frac {21}{2} A c^5 \sin (2 e+f x)-\frac {73}{2} i B c^5 \sin (2 e+f x)\right ) (A+B \tan (e+f x))}{3 f (A \cos (e+f x)+B \sin (e+f x)) (a+i a \tan (e+f x))^2}+\frac {x \sec (e+f x) (\cos (f x)+i \sin (f x))^2 \left (-24 A c^5-56 i B c^5-24 i A c^5 \tan (e)+56 B c^5 \tan (e)+(-3 i A+7 B) \left (8 c^5 \cos (2 e)+8 i c^5 \sin (2 e)\right ) \tan (e)\right ) (A+B \tan (e+f x))}{(A \cos (e+f x)+B \sin (e+f x)) (a+i a \tan (e+f x))^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^5)/(a + I*a*Tan[e + f*x])^2,x]

[Out]

(4*((-3*I)*A + 5*B)*c^5*Cos[2*f*x]*Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])^2*(A + B*Tan[e + f*x]))/(f*(A*Cos[e +
f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + (Sec[e + f*x]*(3*A*c^5*Cos[e] + (7*I)*B*c^5*Cos[e] + (3*I)*
A*c^5*Sin[e] - 7*B*c^5*Sin[e])*(8*ArcTan[Tan[f*x]]*Cos[e] + (8*I)*ArcTan[Tan[f*x]]*Sin[e])*(Cos[f*x] + I*Sin[f
*x])^2*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + (Sec[e + f*x]*(3
*A*c^5*Cos[e] + (7*I)*B*c^5*Cos[e] + (3*I)*A*c^5*Sin[e] - 7*B*c^5*Sin[e])*((4*I)*Cos[e]*Log[Cos[e + f*x]^2] -
4*Log[Cos[e + f*x]^2]*Sin[e])*(Cos[f*x] + I*Sin[f*x])^2*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f
*x])*(a + I*a*Tan[e + f*x])^2) + (Sec[e]*Sec[e + f*x]^3*(3*A*Cos[e] + (21*I)*B*Cos[e] + 2*B*Sin[e])*((I/6)*c^5
*Cos[2*e] - (c^5*Sin[2*e])/6)*(Cos[f*x] + I*Sin[f*x])^2*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f
*x])*(a + I*a*Tan[e + f*x])^2) + ((A + I*B)*Cos[4*f*x]*Sec[e + f*x]*((2*I)*c^5*Cos[2*e] + 2*c^5*Sin[2*e])*(Cos
[f*x] + I*Sin[f*x])^2*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + (
(3*A + (7*I)*B)*Sec[e + f*x]*(8*c^5*f*x*Cos[2*e] + (8*I)*c^5*f*x*Sin[2*e])*(Cos[f*x] + I*Sin[f*x])^2*(A + B*Ta
n[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) - (4*(3*A + (5*I)*B)*c^5*Sec[e + f
*x]*(Cos[f*x] + I*Sin[f*x])^2*Sin[2*f*x]*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*T
an[e + f*x])^2) + ((A + I*B)*Sec[e + f*x]*(2*c^5*Cos[2*e] - (2*I)*c^5*Sin[2*e])*(Cos[f*x] + I*Sin[f*x])^2*Sin[
4*f*x]*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + (Sec[e]*Sec[e +
f*x]^4*(Cos[f*x] + I*Sin[f*x])^2*(-1/2*(B*c^5*Cos[2*e - f*x]) + (B*c^5*Cos[2*e + f*x])/2 - (I/2)*B*c^5*Sin[2*e
- f*x] + (I/2)*B*c^5*Sin[2*e + f*x])*(A + B*Tan[e + f*x]))/(3*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Ta
n[e + f*x])^2) + (Sec[e]*Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*(((-21*I)/2)*A*c^5*Cos[2*e - f*x] + (73*B*c^
5*Cos[2*e - f*x])/2 + ((21*I)/2)*A*c^5*Cos[2*e + f*x] - (73*B*c^5*Cos[2*e + f*x])/2 + (21*A*c^5*Sin[2*e - f*x]
)/2 + ((73*I)/2)*B*c^5*Sin[2*e - f*x] - (21*A*c^5*Sin[2*e + f*x])/2 - ((73*I)/2)*B*c^5*Sin[2*e + f*x])*(A + B*
Tan[e + f*x]))/(3*f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^2) + (x*Sec[e + f*x]*(Cos[f*x] +
I*Sin[f*x])^2*(-24*A*c^5 - (56*I)*B*c^5 - (24*I)*A*c^5*Tan[e] + 56*B*c^5*Tan[e] + ((-3*I)*A + 7*B)*(8*c^5*Cos[
2*e] + (8*I)*c^5*Sin[2*e])*Tan[e])*(A + B*Tan[e + f*x]))/((A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f
*x])^2)

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Maple [A]
time = 0.31, size = 127, normalized size = 0.65

method	result	size
derivativedivides	\(\frac {c^{5} \left (\frac {i B \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {i A \left (\tan ^{2}\left (f x +e \right )\right )}{2}-24 i B \tan \left (f x +e \right )-\frac {7 B \left (\tan ^{2}\left (f x +e \right )\right )}{2}-7 A \tan \left (f x +e \right )-\frac {48 i B +32 A}{-i+\tan \left (f x +e \right )}-\frac {16 i A -16 B}{2 \left (-i+\tan \left (f x +e \right )\right )^{2}}+\left (-24 i A +56 B \right ) \ln \left (-i+\tan \left (f x +e \right )\right )\right )}{f \,a^{2}}\)	\(127\)
default	\(\frac {c^{5} \left (\frac {i B \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {i A \left (\tan ^{2}\left (f x +e \right )\right )}{2}-24 i B \tan \left (f x +e \right )-\frac {7 B \left (\tan ^{2}\left (f x +e \right )\right )}{2}-7 A \tan \left (f x +e \right )-\frac {48 i B +32 A}{-i+\tan \left (f x +e \right )}-\frac {16 i A -16 B}{2 \left (-i+\tan \left (f x +e \right )\right )^{2}}+\left (-24 i A +56 B \right ) \ln \left (-i+\tan \left (f x +e \right )\right )\right )}{f \,a^{2}}\)	\(127\)
risch	\(\frac {20 c^{5} {\mathrm e}^{-2 i \left (f x +e \right )} B}{a^{2} f}-\frac {12 i c^{5} {\mathrm e}^{-2 i \left (f x +e \right )} A}{a^{2} f}-\frac {2 c^{5} {\mathrm e}^{-4 i \left (f x +e \right )} B}{a^{2} f}+\frac {2 i c^{5} {\mathrm e}^{-4 i \left (f x +e \right )} A}{a^{2} f}+\frac {112 i c^{5} B x}{a^{2}}+\frac {48 c^{5} A x}{a^{2}}+\frac {112 i c^{5} B e}{f \,a^{2}}+\frac {48 c^{5} A e}{f \,a^{2}}+\frac {2 c^{5} \left (-18 i A \,{\mathrm e}^{4 i \left (f x +e \right )}+54 B \,{\mathrm e}^{4 i \left (f x +e \right )}-39 i A \,{\mathrm e}^{2 i \left (f x +e \right )}+123 B \,{\mathrm e}^{2 i \left (f x +e \right )}-21 i A +73 B \right )}{3 f \,a^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{3}}-\frac {56 c^{5} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) B}{f \,a^{2}}+\frac {24 i c^{5} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) A}{f \,a^{2}}\)	\(267\)
norman	\(\frac {\frac {-25 i c^{5} A +47 B \,c^{5}}{a f}+\frac {8 \left (7 i c^{5} B +3 A \,c^{5}\right ) x}{a}-\frac {\left (-i c^{5} A +7 B \,c^{5}\right ) \left (\tan ^{6}\left (f x +e \right )\right )}{2 a f}-\frac {7 \left (10 i c^{5} B +3 A \,c^{5}\right ) \left (\tan ^{5}\left (f x +e \right )\right )}{3 a f}+\frac {\left (-83 i c^{5} A +133 B \,c^{5}\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2 a f}+\frac {16 \left (7 i c^{5} B +3 A \,c^{5}\right ) x \left (\tan ^{2}\left (f x +e \right )\right )}{a}+\frac {8 \left (7 i c^{5} B +3 A \,c^{5}\right ) x \left (\tan ^{4}\left (f x +e \right )\right )}{a}-\frac {\left (56 i c^{5} B +23 A \,c^{5}\right ) \tan \left (f x +e \right )}{a f}-\frac {\left (287 i c^{5} B +138 A \,c^{5}\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3 a f}+\frac {i c^{5} B \left (\tan ^{7}\left (f x +e \right )\right )}{3 a f}}{a \left (1+\tan ^{2}\left (f x +e \right )\right )^{2}}+\frac {4 \left (-3 i c^{5} A +7 B \,c^{5}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{a^{2} f}\)	\(318\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^5/(a+I*a*tan(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

1/f*c^5/a^2*(1/3*I*B*tan(f*x+e)^3+1/2*I*A*tan(f*x+e)^2-24*I*B*tan(f*x+e)-7/2*B*tan(f*x+e)^2-7*A*tan(f*x+e)-(48
*I*B+32*A)/(-I+tan(f*x+e))-1/2*(-16*B+16*I*A)/(-I+tan(f*x+e))^2+(-24*I*A+56*B)*ln(-I+tan(f*x+e)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^5/(a+I*a*tan(f*x+e))^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [A]
time = 2.84, size = 343, normalized size = 1.77 \begin {gather*} \frac {2 \, {\left (24 \, {\left (3 \, A + 7 i \, B\right )} c^{5} f x e^{\left (10 i \, f x + 10 i \, e\right )} - 3 \, {\left (3 i \, A - 7 \, B\right )} c^{5} e^{\left (2 i \, f x + 2 i \, e\right )} - 3 \, {\left (-i \, A + B\right )} c^{5} + 12 \, {\left (6 \, {\left (3 \, A + 7 i \, B\right )} c^{5} f x - {\left (3 i \, A - 7 \, B\right )} c^{5}\right )} e^{\left (8 i \, f x + 8 i \, e\right )} + 6 \, {\left (12 \, {\left (3 \, A + 7 i \, B\right )} c^{5} f x - 5 \, {\left (3 i \, A - 7 \, B\right )} c^{5}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + 2 \, {\left (12 \, {\left (3 \, A + 7 i \, B\right )} c^{5} f x - 11 \, {\left (3 i \, A - 7 \, B\right )} c^{5}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 12 \, {\left ({\left (-3 i \, A + 7 \, B\right )} c^{5} e^{\left (10 i \, f x + 10 i \, e\right )} + 3 \, {\left (-3 i \, A + 7 \, B\right )} c^{5} e^{\left (8 i \, f x + 8 i \, e\right )} + 3 \, {\left (-3 i \, A + 7 \, B\right )} c^{5} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-3 i \, A + 7 \, B\right )} c^{5} e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )\right )}}{3 \, {\left (a^{2} f e^{\left (10 i \, f x + 10 i \, e\right )} + 3 \, a^{2} f e^{\left (8 i \, f x + 8 i \, e\right )} + 3 \, a^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^5/(a+I*a*tan(f*x+e))^2,x, algorithm="fricas")

[Out]

2/3*(24*(3*A + 7*I*B)*c^5*f*x*e^(10*I*f*x + 10*I*e) - 3*(3*I*A - 7*B)*c^5*e^(2*I*f*x + 2*I*e) - 3*(-I*A + B)*c
^5 + 12*(6*(3*A + 7*I*B)*c^5*f*x - (3*I*A - 7*B)*c^5)*e^(8*I*f*x + 8*I*e) + 6*(12*(3*A + 7*I*B)*c^5*f*x - 5*(3
*I*A - 7*B)*c^5)*e^(6*I*f*x + 6*I*e) + 2*(12*(3*A + 7*I*B)*c^5*f*x - 11*(3*I*A - 7*B)*c^5)*e^(4*I*f*x + 4*I*e)
- 12*((-3*I*A + 7*B)*c^5*e^(10*I*f*x + 10*I*e) + 3*(-3*I*A + 7*B)*c^5*e^(8*I*f*x + 8*I*e) + 3*(-3*I*A + 7*B)*
c^5*e^(6*I*f*x + 6*I*e) + (-3*I*A + 7*B)*c^5*e^(4*I*f*x + 4*I*e))*log(e^(2*I*f*x + 2*I*e) + 1))/(a^2*f*e^(10*I
*f*x + 10*I*e) + 3*a^2*f*e^(8*I*f*x + 8*I*e) + 3*a^2*f*e^(6*I*f*x + 6*I*e) + a^2*f*e^(4*I*f*x + 4*I*e))

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Sympy [A]
time = 0.82, size = 445, normalized size = 2.29 \begin {gather*} \frac {- 42 i A c^{5} + 146 B c^{5} + \left (- 78 i A c^{5} e^{2 i e} + 246 B c^{5} e^{2 i e}\right ) e^{2 i f x} + \left (- 36 i A c^{5} e^{4 i e} + 108 B c^{5} e^{4 i e}\right ) e^{4 i f x}}{3 a^{2} f e^{6 i e} e^{6 i f x} + 9 a^{2} f e^{4 i e} e^{4 i f x} + 9 a^{2} f e^{2 i e} e^{2 i f x} + 3 a^{2} f} + \begin {cases} \frac {\left (\left (2 i A a^{2} c^{5} f e^{2 i e} - 2 B a^{2} c^{5} f e^{2 i e}\right ) e^{- 4 i f x} + \left (- 12 i A a^{2} c^{5} f e^{4 i e} + 20 B a^{2} c^{5} f e^{4 i e}\right ) e^{- 2 i f x}\right ) e^{- 6 i e}}{a^{4} f^{2}} & \text {for}\: a^{4} f^{2} e^{6 i e} \neq 0 \\x \left (- \frac {48 A c^{5} + 112 i B c^{5}}{a^{2}} + \frac {\left (48 A c^{5} e^{4 i e} - 24 A c^{5} e^{2 i e} + 8 A c^{5} + 112 i B c^{5} e^{4 i e} - 40 i B c^{5} e^{2 i e} + 8 i B c^{5}\right ) e^{- 4 i e}}{a^{2}}\right ) & \text {otherwise} \end {cases} + \frac {8 i c^{5} \cdot \left (3 A + 7 i B\right ) \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{a^{2} f} + \frac {x \left (48 A c^{5} + 112 i B c^{5}\right )}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))**5/(a+I*a*tan(f*x+e))**2,x)

[Out]

(-42*I*A*c**5 + 146*B*c**5 + (-78*I*A*c**5*exp(2*I*e) + 246*B*c**5*exp(2*I*e))*exp(2*I*f*x) + (-36*I*A*c**5*ex
p(4*I*e) + 108*B*c**5*exp(4*I*e))*exp(4*I*f*x))/(3*a**2*f*exp(6*I*e)*exp(6*I*f*x) + 9*a**2*f*exp(4*I*e)*exp(4*
I*f*x) + 9*a**2*f*exp(2*I*e)*exp(2*I*f*x) + 3*a**2*f) + Piecewise((((2*I*A*a**2*c**5*f*exp(2*I*e) - 2*B*a**2*c
**5*f*exp(2*I*e))*exp(-4*I*f*x) + (-12*I*A*a**2*c**5*f*exp(4*I*e) + 20*B*a**2*c**5*f*exp(4*I*e))*exp(-2*I*f*x)
)*exp(-6*I*e)/(a**4*f**2), Ne(a**4*f**2*exp(6*I*e), 0)), (x*(-(48*A*c**5 + 112*I*B*c**5)/a**2 + (48*A*c**5*exp
(4*I*e) - 24*A*c**5*exp(2*I*e) + 8*A*c**5 + 112*I*B*c**5*exp(4*I*e) - 40*I*B*c**5*exp(2*I*e) + 8*I*B*c**5)*exp
(-4*I*e)/a**2), True)) + 8*I*c**5*(3*A + 7*I*B)*log(exp(2*I*f*x) + exp(-2*I*e))/(a**2*f) + x*(48*A*c**5 + 112*
I*B*c**5)/a**2

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (172) = 344\).
time = 1.07, size = 517, normalized size = 2.66 \begin {gather*} \frac {2 \, {\left (\frac {12 \, {\left (3 i \, A c^{5} - 7 \, B c^{5}\right )} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2}} - \frac {24 \, {\left (3 i \, A c^{5} - 7 \, B c^{5}\right )} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}{a^{2}} - \frac {12 \, {\left (-3 i \, A c^{5} + 7 \, B c^{5}\right )} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}{a^{2}} - \frac {66 i \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 154 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 21 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 72 i \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 201 i \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 483 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 42 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 148 i \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 201 i \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 483 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 21 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 72 i \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 66 i \, A c^{5} + 154 \, B c^{5}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3} a^{2}} - \frac {2 \, {\left (-75 i \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 175 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 324 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 748 i \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 522 i \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1170 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 324 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 748 i \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 75 i \, A c^{5} + 175 \, B c^{5}\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}^{4}}\right )}}{3 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^5/(a+I*a*tan(f*x+e))^2,x, algorithm="giac")

[Out]

2/3*(12*(3*I*A*c^5 - 7*B*c^5)*log(tan(1/2*f*x + 1/2*e) + 1)/a^2 - 24*(3*I*A*c^5 - 7*B*c^5)*log(tan(1/2*f*x + 1
/2*e) - I)/a^2 - 12*(-3*I*A*c^5 + 7*B*c^5)*log(tan(1/2*f*x + 1/2*e) - 1)/a^2 - (66*I*A*c^5*tan(1/2*f*x + 1/2*e
)^6 - 154*B*c^5*tan(1/2*f*x + 1/2*e)^6 - 21*A*c^5*tan(1/2*f*x + 1/2*e)^5 - 72*I*B*c^5*tan(1/2*f*x + 1/2*e)^5 -
201*I*A*c^5*tan(1/2*f*x + 1/2*e)^4 + 483*B*c^5*tan(1/2*f*x + 1/2*e)^4 + 42*A*c^5*tan(1/2*f*x + 1/2*e)^3 + 148
*I*B*c^5*tan(1/2*f*x + 1/2*e)^3 + 201*I*A*c^5*tan(1/2*f*x + 1/2*e)^2 - 483*B*c^5*tan(1/2*f*x + 1/2*e)^2 - 21*A
*c^5*tan(1/2*f*x + 1/2*e) - 72*I*B*c^5*tan(1/2*f*x + 1/2*e) - 66*I*A*c^5 + 154*B*c^5)/((tan(1/2*f*x + 1/2*e)^2
- 1)^3*a^2) - 2*(-75*I*A*c^5*tan(1/2*f*x + 1/2*e)^4 + 175*B*c^5*tan(1/2*f*x + 1/2*e)^4 - 324*A*c^5*tan(1/2*f*
x + 1/2*e)^3 - 748*I*B*c^5*tan(1/2*f*x + 1/2*e)^3 + 522*I*A*c^5*tan(1/2*f*x + 1/2*e)^2 - 1170*B*c^5*tan(1/2*f*
x + 1/2*e)^2 + 324*A*c^5*tan(1/2*f*x + 1/2*e) + 748*I*B*c^5*tan(1/2*f*x + 1/2*e) - 75*I*A*c^5 + 175*B*c^5)/(a^
2*(tan(1/2*f*x + 1/2*e) - I)^4))/f

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Mupad [B]
time = 8.81, size = 282, normalized size = 1.45 \begin {gather*} -\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (-\frac {56\,B\,c^5}{a^2}+\frac {A\,c^5\,24{}\mathrm {i}}{a^2}\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (-\frac {3\,B\,c^5}{2\,a^2}+\frac {c^5\,\left (A+B\,4{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,a^2}\right )}{f}-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (\frac {3\,c^5\,\left (A+B\,4{}\mathrm {i}\right )}{a^2}+\frac {B\,c^5\,6{}\mathrm {i}}{a^2}-\frac {c^5\,\left (-3\,B+A\,2{}\mathrm {i}\right )\,2{}\mathrm {i}}{a^2}\right )}{f}+\frac {-\frac {\left (-24\,B\,c^5+A\,c^5\,8{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,a^2}+\frac {16\,A\,c^5+B\,c^5\,64{}\mathrm {i}}{2\,a^2}+\frac {\left (-56\,B\,c^5+A\,c^5\,24{}\mathrm {i}\right )\,3{}\mathrm {i}}{2\,a^2}+\mathrm {tan}\left (e+f\,x\right )\,\left (\frac {\left (16\,A\,c^5+B\,c^5\,64{}\mathrm {i}\right )\,1{}\mathrm {i}}{a^2}-\frac {2\,\left (-56\,B\,c^5+A\,c^5\,24{}\mathrm {i}\right )}{a^2}\right )}{f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^2\,1{}\mathrm {i}+2\,\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )}+\frac {B\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^3\,1{}\mathrm {i}}{3\,a^2\,f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*tan(e + f*x))*(c - c*tan(e + f*x)*1i)^5)/(a + a*tan(e + f*x)*1i)^2,x)

[Out]

(tan(e + f*x)^2*((c^5*(A + B*4i)*1i)/(2*a^2) - (3*B*c^5)/(2*a^2)))/f - (log(tan(e + f*x) - 1i)*((A*c^5*24i)/a^
2 - (56*B*c^5)/a^2))/f - (tan(e + f*x)*((3*c^5*(A + B*4i))/a^2 + (B*c^5*6i)/a^2 - (c^5*(A*2i - 3*B)*2i)/a^2))/
f + ((16*A*c^5 + B*c^5*64i)/(2*a^2) - ((A*c^5*8i - 24*B*c^5)*1i)/(2*a^2) + ((A*c^5*24i - 56*B*c^5)*3i)/(2*a^2)
+ tan(e + f*x)*(((16*A*c^5 + B*c^5*64i)*1i)/a^2 - (2*(A*c^5*24i - 56*B*c^5))/a^2))/(f*(2*tan(e + f*x) + tan(e
+ f*x)^2*1i - 1i)) + (B*c^5*tan(e + f*x)^3*1i)/(3*a^2*f)

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